Flow-Induced Motion and Energy Conversion of the Cir-T-Att Oscillator in a Flow Field with a High Reynolds Number

Abstract

The present study aims to systematically investigate the effects of a high Reynolds number on the flow-induced motion and energy conversion of the Cir-T-Att oscillator. Experiments are conducted in six Reynolds number ranges (2.89 × 10⁴~6.51 × 10⁴ < Re < 7.71 × 10⁴~15.21 × 10⁴) in regimes of TrSL2, TrSL3 and TrBL0. The system total damping ratio (ζ_total) is adjusted by varying excitation voltages (V_B) with a controllable magnetic damping system. The V_B is varied from 0 V to 165 V, corresponding to 0.082 ≤ ζ_total ≤ 1.153. The amplitude, frequency, fluid force, spectral content, active power, and upper limit of power output are analyzed. The results show that the Reynolds number affects both amplitude and global response characteristics. The active power increases with an increasing Reynolds number within the upper branch. The maximum power output of the Cir-T-Att oscillator reaches 10.43 W, appearing at Re = 14.67 × 10⁴ (D = 0.16 m, ζ_total = 0.468, U_r = 6.34), while the maximum upper limit of power output is 17.80 W at Re = 15.21 × 10⁴ (D = 0.16 m, ζ_total = 0.678, U_r = 6.57).

1. Introduction

In recent years, to realize carbon neutrality, energy savings, and emission reduction have remained a critical research hotspot [1]. Researchers aim at harvesting green energy from nature so as to achieve the adjustment of energy structure [2]. However, as a renewable resource with abundant reserves in the environment, water resources still have great development value [3,4,5,6,7], such as harvesting energy from flow induced motions [8]. Previous research has focused on adopting oscillators with various sections [9,10,11,12], employing mechanical non-linearities [13], or basing on different energy conversion principles (electromagnetic [14], piezoelectric [15], and so on) to improve the energy conversion ability [16]. The relevant reviews systematically introduced more developments [2,17].

Despite large progress on the flow-induced motion energy conversion system (FIMECS), few researchers have systematically studied the influence of the Reynolds number (Re = ρ·U·D/μ, ρ is the fluid density, U is the current velocity, D is the characteristic width, and μ is the viscosity) on energy conversion in laboratory tests. Over a long period of time, due to the limitations of material strength and test equipment condition, studies on flow-induced motion at a high Reynolds number have been very scarce, with research mainly measuring the Strouhal number of the stationary cylinder [17]. The influence of Reynolds number on flow-induced vibration of a cylinder is regarded as weak in early research. As reviewed by Govardhan and Williamson [18], the effect of Reynolds number on vibration response has been revealed by many scholars at a low Reynolds number regime. Additionally, many researchers have carried out a lot of research on flow-induced vibration of cylinders in the high Reynolds number range during the past decade [19,20,21] and found that the Reynolds number range has a significant impact on the range and amplitude of the synchronous range. In addition, Prassath et al. [22] applied a cartesian grid to simulate the wake flow features of a triangular prism with different slenderness ratios and different directions of incoming flow under a low Reynolds number regime and successfully predicted the critical Reynolds numbers when the triangular prism began to shed a vortex under different conditions. Subsequently, Tu et al. [23] have simulated the wake patterns of stationary regular triangular prism under different Reynolds numbers and different attack angles; also, the wake patterns of rotating regular prism under different attack angles, Reynolds numbers, aptitudes, and rotation frequencies are investigated, and the Reynolds number affects the wake morphology and lift coefficient of the prism significantly. The effect of Reynolds number on the flow-induced motion of the non-circular cylinders and the rounded flowing pattern is significant.

According to the fluid characteristics of the separated shear layer and the near-surface boundary layer of the cylinder, Zdravkovich et al. [24] and Bearman et al. [25] classified the flow pattern around the smooth cylinder into 15 flow zones. When 1 × 10³ < Re < 1 × 10⁴ (TrSL1, TrSL2), the lift coefficient of the cylinder is small, but with the increase of Re, the lift coefficient increases rapidly. When 1 × 10⁴ < Re < 1 × 10⁵, the wake flow pattern is located at the end of TrSL2, and in TrSL3, a platform appears in the lift curve of the cylinder at this time. When Re increases to 1 × 10⁵ ~ 1 × 10⁶, the wake flow pattern is located in the TrBL regime, and the lift curve of the cylinder changes sharply, which is corresponding to the transition of the boundary layer, and the turbulent boundary layer increases the stress in the wake. As noticed by Unal et al. [26], the lifting force on the cylinder is closely related to the magnitude of the velocity fluctuation in the shear layer. Additionally, the wake flow characteristics of the cylinder are affected by the roughness of the cylinder surface [27]. As seen from Chang et al. [28] and Park et al. [29], the boundary layer transition can be effectively triggered by installing rough attachments on the symmetric position on the surface of a smooth cylinder, that is, in the range of subcritical (laminar) Reynolds numbers, the fluid will show the flow characteristics of supercritical (turbulent) flow.

As summarized above, it can be seen that both the flow-induced motion of circular cylinders and non-circular cylinders are sensitive to the effect of Reynolds number; the summary of a representative investigation of Reynolds number effect on vortex induced vibration is summarized in

Table 1. Representative studies.

Reference	Cylinders	Re	Working Fluid	Experiment/ Numeric	Results
[19]	stationary	106–107	wind	experiment	St = 0.27
[31]	flexible	190	wind	theory	A* = 0.5
[20]	elastic supports	4.4 × 104–13.4 × 105	water	experiment	ηPeak = 0.308
[32]	flexible	8 × 104–1 × 106	water	numeric	A* = 0.9
[3]	elastic supports	96–118	water	experiment	A* = 0.325

. So far, the studies of the flow regime around a circular cylinder are relatively complete at different Reynolds numbers, while the understanding of the wake pattern and vibration response of noncylinders in different Re number ranges is far from systematic and in-depth. Meanwhile, in our previous studies on different combined cross sections [30], we found that the Cir-T-Att oscillator has a great advantage in flow-induced motion energy harvesting, so this study is related to the Cir-T-Att oscillator under different Reynolds numbers. In the present study, experiments are conducted in a recirculating water channel, and the controllable magnetic damping system is applied to examine the effect of system damping on the output power and system efficiency under different Reynolds numbers.

In the present study, the experimental setup and the mathematical relations of the FIMECS are presented in Section 2. In Section 2.3, the validation of the physical model is briefly described. Section 3 illustrates the results and discussion of FIMECS under different Reynolds number ranges. In Section 3.1, the response of amplitude and frequency under different Reynolds numbers is presented and discussed. Section 3.2 and Section 3.3, respectively, give the effects of the Reynolds number on the fluid force and spectral content of the Cir-T-Att oscillator. Additionally, the active power and the upper limit of power output are analyzed to reveal the Reynolds number effect on the levels of the harvested power in Section 3.4 and Section 3.5. Finally, the main conclusions are mentioned in Section 4.

2. Experimental Setup

2.1. Test Apparatus and Method

The experiment was conducted in the State Key Laboratory in the Hydraulic Engineering Simulation and Safety Testing Hall at Tianjin University. As shown in Figure 1, the test system is composed of an open horizontal circulation channel, a supplement tank, the power part, the control section, and the test device of flow-induced motion and energy conversion system. The total length of the flow channel is about 50 m. It has a 2-m-wide channel section, a curved channel section, a contraction channel section, and a 1-m-wide channel section along the flow direction. At the same time, the channel wall is made of transparent glass, which allows for complete visualization of the flow and test section. The power part applies a large frequency conversion tubular pump to promote water movement, and the power of the power pump can be up to 90 kW. The size of the makeshift water tank is 5 m × 5 m × 2 m, with a water storage capacity of about 200 m³. The control box mainly uses the frequency converter to adjust the pump speed and control the outflow speed. The frequency conversion range of the frequency converter is 0.0~50.0 Hz. The test device for flow-induced motion and the energy conversion system is arranged in a 1-m-wide channel section, the water depth is controlled at 1.34 m, and the incoming flow velocity is 0~2 m/s.

The principle of FIMECS is shown in Figure 2. It is composed of an oscillation part, a transmission part, and a power take-off part [33]. The model of FIMECS considered in this study includes the coupling effects of water flow, generator, and resistance. We simplified it as a mass-stiffness-damping model. The incoming fluid flows vertically to the oscillator, and because the oscillator is subject to the lateral limit, it can only vibrate transversely with a single degree of freedom under the action of the flow force. Then its motion can be approximated by the linear dynamic Equation (1). Then, the movement of the oscillator drives the rotor movement of the generator through the transmission part, and the kinetic energy of the fluid is converted into electrical energy. Moreover, the magnetic flux density of the generator can be changed by an external excitation voltage. Finally, the generated electric energy is supplied to the internal R₀ loss of the generator and the power consumption of the external load R_L.

$$\left(m_{\mathrm{osc}}+m_{\mathrm{a}}\right)\ddot{y}+c_{\mathrm{total}}\dot{y}+Ky=F_{\mathrm{L}}$$

(1)

where m_osc is the oscillation mass, m_a is the added water mass, and y, $\dot{y}$, and $\ddot{y}$ are the displacement, velocity, and acceleration of the oscillator, respectively. c_total is the total damping, K is the system stiffness, and F_L is the fluid force in the transverse direction.

During the test, the displacement of the oscillator is measured by a magnetic induction displacement transducer; the testing range is 0–800 mm; the sensitivity is 0.1%; and the accuracy is ±0.05% under full scale. The amplitude A is the average of all amplitudes in a time series of displacements measured in 60 s. The amplitude ratio A* is the ratio of amplitude and diameter (A* = A/D), and the maximum error of A* is ±0.004. The f* is the frequency ratio (f* = f_osc/f_n,air), f_osc is the main frequency of oscillation obtained from the displacement time-history curves by the FFT method, and f_n,air is the natural frequency in air.

To calculate the lift coefficient c_L(t), the torque at the end of the transmission part is measured by the HCNJ-103 torque sensor. The testing range is 0~2 Nm, the accuracy is 0.5%, and the maximum error of F_T is less than ±0.3333 N. The c_L(t) is shown in Equation (2).

$$c_{\mathrm{L}}(t)=\frac{2\left[(m_{\mathrm{osc}}+m_{\mathrm{a}})\ddot{y}+c\dot{y}+Ky-\frac{T(t)}{R_{\mathrm{m}}}\right]}{\rho U^{2}Dl}$$

(2)

where T(t) and R_m (0.03 m) are the torque and the gear radius at the end of the transmission part, respectively, and c is the mechanical damping.

The active power is calculated by Equation (3), and the upper limit of power for FIMECS can be expressed by Equation (5). The voltage variation of the external resistance is collected in real time; the measurement range is from −10 V to +10 V with accuracy within ±0.1% FS. The maximum error of P_harn is less than ±0.0026 W.

$$P_{\mathrm{harn}}(t)=\frac{u^2(t)}{R_{\mathrm{L}}}$$

(3)

$$P_{\mathrm{mech}}=c\cdot {\dot{y}}^2$$

(4)

$$P_{\mathrm{UL}}=P_{\mathrm{mech}}+P_{\mathrm{harn}}$$

(5)

2.2. Test Condition

In the present study, we focus on investigating the influence of Reynolds number on the FIMECS with the Cir-T-Att oscillator; therefore, the total system mass, the system stiffness, and the length L of the Cir-T-Att oscillator in different Re ranges are controlled to be the same. Referring to the methods of previous representative studies [21], different Reynolds numbers are achieved by changing the characteristic width of the oscillator D. The oscillator model is shown in Figure 3, and the relevant parameters are shown in

Table 2. Test condition.

D/m	L/m	H/m	K/(N/m)	RL/Ω	U/(m/s)	Re	Regime
0.06	0.9	0.06	1860	26	0.55~1.24	28.9 k~65.1 k	TrSL2, TrSL3
0.08	0.9	0.08	1860	26	0.55~1.24	38.6 k~86.7 k	TrSL2, TrSL3
0.10	0.9	0.10	1860	26	0.55~1.08	48.2 k~95.0 k	TrSL3
0.12	0.9	0.12	1860	26	0.55~1.08	57.9 k~114.1 k	TrSL3, TrBL0
0.14	0.9	0.14	1860	26	0.55~1.08	67.5 k~133.1 k	TrSL3, TrBL0
0.16	0.9	0.16	1860	26	0.55~1.08	77.1 k~152.1 k	TrSL3, TrBL0

.

All oscillators were 0.9 m long and made of polymethyl methacrylate. Considering the boundary effect zone, two end plates are set at both ends of the oscillator, and the thickness of the end plate is designed as δ = 0.01 m. Model diameters are 0.06, 0.08, 0.10, 0.12, 0.14, and 0.16 m. As referred to in the flow regime classification by Zdravkovich et al. [25], the test range is in the TrSL2, TrSL3, and TrBL0 regimes.

2.3. Validation of Physical Model

In this section, repeated free decay tests are conducted to obtain the natural frequencies of FIMECS in air and water under different system damping. To ensure the natural frequencies of FIMECS with different characteristic width oscillators are identical, various weights are added or reduced to the FIMECS. The total damping ratio ζ_total, the oscillation mass m_osc, the additional water mass ma, and the total damping c_total are derived as Equations (6)–(9), respectively. The results of the free decay test in air for different characteristic width oscillators are similar, so the detailed results of D = 0.06 m are listed in

Table 3. Results of free decay tests in the air (D = 0.06 m).

VB/V	fn,a/Hz	mosc/kg	ζtotal	ctotal(N·s·m−1)
0.00	1.03	44.53	0.082	47.068
3.00	1.03	43.97	0.087	49.483
9.00	1.03	44.53	0.090	51.644
15.00	1.03	43.97	0.093	52.861
18.00	1.03	43.64	0.096	54.303
21.00	1.03	43.97	0.112	64.129
27.00	1.03	44.55	0.122	70.363
36.00	1.03	43.64	0.134	76.008
42.00	1.03	43.63	0.152	86.46
45.00	1.05	42.29	0.172	96.244
51.00	1.06	41.75	0.197	109.649
57.00	1.05	42.28	0.217	121.498
63.00	1.04	43.44	0.246	139.327
69.00	1.03	43.99	0.242	137.663
75.00	1.03	43.64	0.304	172.674
81.00	1.03	43.64	0.341	193.575
87.00	1.03	43.64	0.381	216.084
93.00	1.03	43.64	0.423	240.200
99.00	1.03	43.64	0.468	265.924
105.00	1.03	43.64	0.516	293.256
111.00	1.02	43.64	0.567	322.196
117.00	1.05	43.64	0.621	352.743
123.00	1.03	43.64	0.678	384.90
129	1.03	43.64	0.737	418.66
165	1.03	43.64	1.153	655.00

as representative. The results of free decay tests in the water are listed in

Table 4. Results of free decay tests in the water.

D/m	0.06	0.08	0.10	0.12	0.14	0.16
fn,w/Hz	0.96	0.95	0.92	0.89	0.84	0.79
(ma + mosc)/kg	50.35	52.27	55.41	58.97	66.77	74.34

. The results illustrate that f_n,a and m_osc almost remain unchanged under different system damping, with m_osc = 43.635 kg, f_n,a = 1.03 Hz, and the mechanical damping coefficient c = 47.068 N·s/m.

$${\zeta }_{\mathrm{total}}=\frac{\ln \eta }{2\pi }=\frac{1}{2\pi }\ln \left(\frac{A_{\mathrm{i}}}{A_{\mathrm{i}+1}}\right)$$

(6)

$$m_{\mathrm{osc}}=\frac{K}{(2\pi f_{\mathrm{n},\mathrm{a}}{)}^2}$$

(7)

$$m_{\mathrm{a}}+m_{\mathrm{osc}}=\frac{K}{(2\pi f_{\mathrm{n},\mathrm{w}}{)}^2}$$

(8)

$$c_{\mathrm{total}}=2\sqrt{m_{\mathrm{osc}}K}\cdot {\zeta }_{\mathrm{total}}$$

(9)

where f_n,w is the natural frequency of the oscillator in water.

The calculated total damping c_{total-calculated} is calculated by (Equation (10)).

$$c_{\mathrm{total}-\mathrm{caculated}}=47.068+0.02233{V_{\mathrm{B}}}^2$$

(10)

The error between the calculated total damping and the tested total damping is shown in Figure 4. The physical model is in good agreement with the mathematical relations, with almost no deviations higher than 5%. The reason for the deviations between c_total-tested and c_{total-calculated} may be the error of the instrument.

3. Results and Discussion

Five aspects of energy conversion from flow induced motion are studied to evaluate the effect of Reynolds number on them in this part. To clearly observe rules, we select only 5 typical curves for each oscillator that are not under adjacent conditions to illustrate. On account of oscillation stability and intensity for an oscillator with different characteristic widths, the upper limit of the total damping ratio ζ_total and the current velocity U were reasonably adjusted.

3.1. Influence of Re on Oscillation Response

As shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, A* and f* versus current velocity (U), reduced velocity (U_r), and Reynolds number (Re) under different ζ_total values for each oscillator are presented. For D = 0.06 m, when ζ_total over 0.122, the oscillation cannot keep itself stable. While for D = 0.08 m, D = 0.10 m, D = 0.12 m, D = 0.14 m, and D = 0.16 m, the maximum system total damping ζ_total,max of different oscillators can overcome is 0.152, 0.246, 0.423, 1.158, and 0.678 respectively. It is noticed that the ζ_total,max increases with an increase the Reynolds number from D = 0.06 m to D = 0.14 m. However, when D is over 0.14, the ζ_total,max decreases, and it may be that the large characteristic width causes an excessive blocking ratio, which leads to the water surface being largely deformed by the oscillator and the oscillation stability and intensity are affected by it. In this study, the maximum system total damping ζ_total,max of different oscillators can overcome is considered the maximum ζ_total until the oscillation cannot keep itself stable.

In Figure 5, the current velocity U is from 0.55 m/s to 1.24 m/s, the reduced velocity U_r ranges from 8.89 to 19.98, and the corresponding Reynolds number Re varies from 2.89 × 10⁴ to 6.51 × 10⁴ for D = 0.06 m. In this Reynolds number range, as ζ_total ≤ 0.122, A^* overall trend continues to increase, while f^* tends to stabilize at about 0.9. With the increase in Reynolds number, the oscillator successively enters VIV branches and VIV-Galloping transition branches, but galloping branches are not fully developed. The maximum amplitude ratio A*_max is 2.02 and appears at Re = 6.51 × 10⁴ (U_r = 19.98) with ζ_total = 0.087.

As shown in Figure 6, the current velocity U is from 0.55 m/s to 1.24 m/s, the reduced velocity U_r ranges from 6.67 to 14.99, and the corresponding Reynolds number Re varies from 3.86 × 10⁴ to 8.67 × 10⁴ for D = 0.08 m. When ζ_total < 0.134, the oscillation undergoes an SG trend, but galloping branches are not clearly observed. As Re increases, A^* first increases and then tends to stabilize with a slight increase, and f^* levels at approximately 0.9, indicating that the oscillator is in VIV initial branches and VIV upper branches. While the oscillation undergoes a HG trend when 0.134 ≤ ζ_total ≤ 0.152, as Re increases, the oscillator enters VIV initial branches, VIV upper branches, and VIV lower branches in turn. However, the VIV lower branches can be induced to galloping branches with an external excitation. The maximum amplitude ratio A^*_max is 1.81, appearing at Re = 8.67 × 10⁴ (U_r = 14.99) with ζ_total = 0.087. In Figure 7, the current velocity U is from 0.55 m/s to 1.08 m/s, the reduced velocity U_r ranges from 5.33 to 10.51, and the corresponding Reynolds number Re varies from 4.82×10⁴ to 9.50 × 10⁴ for D = 0.10 m. The oscillation response is similar to that of D = 0.08 m, but when ζ_total exceeds 0.246, the VIV lower branches cannot be pushed into the galloping branch. The maximum amplitude ratio A^*_max is 1.86, appearing at Re = 9.5 × 10⁴ (U_r = 10.51) with ζ_total = 0.087.

Then, for D = 0.12 m, as shown in Figure 8, the current velocity U is from 0.55 m/s to 1.08 m/s, the reduced velocity U_r varies from 4.44 to 8.76, and the corresponding Reynolds number ranges from 5.79 × 10⁴ to 11.41 × 10⁴. With the increase of Re, A^* first increases, then tends to sable with a slight fluctuation, and f^* also shows an increase, then tends to sable around 0.9, indicating that the oscillation is in the VIV branch. The maximum amplitude ratio A^*_max is 1.73, appearing at Re = 11.41 × 10⁴ (U_r = 8.76) with ζ_total = 0.087. As shown in Figure 9 and Figure 10, the oscillation character of D = 0.14 m and D = 0.16 m is similar to that of D = 0.12 m. For D = 0.14 m, the current velocity U is ranges from 0.55 m/s to 1.08 m/s, the reduced velocity U_r varies from 3.81 to 7.51 and corresponding Reynolds number from 6.75 × 10⁴ to 13.31 × 10⁴, its A^*_max is 1.71 appears at Re = 13.31 × 10⁴ (U_r = 7.51) with ζ_total = 0.087. While for D = 0.16 m, the current velocity U ranges from 0.55 m/s to 1.08 m/s, the reduced velocity U_r varies from 3.33 to 6.57 and the corresponding Reynolds number from 7.71 × 10⁴ to 15.21 × 10⁴, its A^*_max is 1.41 and appears at Re = 15.21 × 10⁴ (U_r = 6.57) with ζ_total = 0.087. From our experiments, the Reynolds number has a strong influence on the oscillation response; as Re increases, the oscillation intensity and the VIV upper branch range increase.

3.2. Effect of Re on Fluid Force

As shown in Figure 11a–f, variations of the lift coefficient C_L with the Reynolds number Re and total damping ratio ζ_total are illustrated. For D = 0.06 m, C_L shows an increasing trend with the increase of Re in the VIV branches. As Re keeps on increasing, C_L shows a decreasing trend in the VIV-Galloping transition branches. It may be because, with the increase in attack angle α, C_L increases before the fluid boundary layer is separated. The angle of attack continues to increase, until it exceeds the critical value.

However, when α continues to increase until the fluid boundary layer separates, C_L shows a decreasing trend with the increase of Re, leading to oscillator stall, as will be further discussed in the spectrum part. For D = 0.08 m, the oscillation undergoes the SG trend when ζ_total ≤ 0.134, C_L first tends to stabilize along with a slightly increasing trend, then decreases, indicating that the oscillation is transitioning from the VIV upper branch to the VIV-Galloping transition branches. However, the oscillation undergoes a HG trend when 0.134 ≤ ζ_total ≤ 0.152, C_L suddenly collapses to 0 when the oscillation undergoes into the VIV lower branches. It may be because in the VIV lower branches, the oscillation cannot induce or keep vortex shedding stable.

Additionally, for D = 0.10 m, the C_L character is similar to that of D = 0.08 m. However, the C_L first increases, then tends to stabilize along with a slightly increasing trend, finally decreasing, which compared with D = 0.08 m, contains the initial branch. This inconsistency may be because the starting reduced velocity of D = 0.10 is lower than that of D = 0.08 m as Re increases. For D = 0.12 m, when U_r is relatively small, the C_L shows a fluctuating trend. This may be because the vortices and vertex shedding cannot generate, and the force on the oscillator surface is unstable. As U_r continues to increase, the C_L increases steadily, which is because the oscillator is only driven by the lift caused by vortex shedding in the VIV branches. The performance of D = 0.14 m and D = 0.16 m is similar to that of D = 0.12 m, and the lift coefficients strongly depend on reduced velocity in the TrSL3 and TrBL0 flow regimes (D = 0.12 m, D = 0.14 m, and D = 0.16 m).

3.3. Spectral Content Dependence on Re

To study the influence of Re on the growth of harmonic components, the displacement and fluid force spectra of the system damping ratio ζ_total = 0.087 are discussed in detail as a typical example. As shown in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, compared with the higher harmonic component in the force spectrum, that of the displacement spectrum has less influence. Higher harmonics are observed at higher Re, and the high amplitudes occurred in the VIV upper branch, which may be because the wake structure is restructuring itself.

The spectral content of six oscillators under three different reduced velocities U_r is observed: For D = 0.06 m, the Re = 2.89 × 10⁴ (U_r = 8.89) is in the VIV branch. There is a prominent peak (f_osc = 0.909 Hz) at the first harmonic component of the oscillation frequency. Additionally, in the fluid force spectra, two peaks, f_L = 0.929 Hz and f_L = 2.828 Hz, are observed at the first harmonic component and the third harmonic component, respectively. In the second range, the Re = 3.70 × 10⁴ (U_r = 11.35) is also referring to the range of the VIV branch. In displacement spectra, only a prominent peak (f_osc = 1.005 Hz) is observed at the first harmonic component of the oscillation frequency. There are two peaks, f_L = 1.005 Hz and f_L = 2.010 Hz, at the first harmonic component and the second harmonic component, respectively. These results corroborate the ideas of Raghavan [21]. In the VIV branch, the main frequency of oscillation is obvious and there is no higher harmonic, which indicates that the fluid is vortex shedding in 2S mode, that is, there is a vortex shedding at the position where the oscillator runs to the maximum displacement of positive and negative. Its force mechanism is that the periodic lifting force caused by vortex shedding drives the oscillator to oscillate.

In the third range, Re = 6.51 × 10⁴ (U_r = 19.98), the following observations are made: a prominent peak (f_osc = 0.935 Hz) and a small peak (f_osc = 1.870) in the displacement spectra are observed, and there are 4 peaks (f_L = 0.935 Hz, f_L = 1.870 Hz, f_L = 2.788 Hz, and f_L = 3.740 Hz) in the force spectra. Referring to the results of Ding et al. and Zhang et al. [34,35] and combining with the spectra curve obtained from the test, the wake vortex shedding mode of the oscillation in the range before jumping from the VIV upper branch to the galloping branch is inferred. In the displacement spectra, the second and third harmonic components of the oscillation frequency are observed, and the vortex structure is in the combination mode of 2P and P + S or the combination mode of 2P and 2S.

For the case of D = 0.08 m, when Re = 3.86 × 10⁴ (U_r = 6.67), the oscillator undergoes a VIV response; a prominent peak is f_osc = 0.808 in displacement spectra, and there are 2 peaks (f_L = 0.808 Hz and f_L = 3.737 Hz) in the force spectra, so the vortex mode is 2S. For Re = 6.00 × 10⁴ (U_r = 10.36), the oscillator is still undergoing a VIV response; only a peak (f_osc = 0.935) is observed at the primary oscillating frequency, and there are 3 peaks (f_L = 0.935 Hz, f_L = 1.886 Hz, and f_L = 2.821 Hz) in the force spectra, the vortex structures are also 2S mode. When Re = 8.67 × 10⁴ (U_r = 14.99), the spectra have two prominent peaks f_osc = 0.918 and f_osc = 0.935, the oscillation is in the transition branch. The oscillation mechanism is that the oscillator oscillates under the dual effects of vortex shedding and lift instability. While 3 peaks (f_L = 0.918 Hz, f_L = 1.836 Hz, and f_L = 2.755 Hz) are observed in the force spectra.

As D = 0.10 m, the following observations are made: when Re = 4.82 × 10⁴ (U_r = 5.33), the displacement spectra only have a peak (f_osc = 0.808), and a prominent peak is observed in the force spectra (f_L = 0.808 Hz), the vortex mode is 2S in the VIV branch. As Re = 6.83 × 10⁴ (U_r = 7.55), the oscillator is still undergoing a VIV response; a conspicuous peak (f_osc = 0.868) is observed in the displacement spectra, and 3 peaks (f_L = 0.868 Hz, f_L = 2.588 Hz, and f_L = 4.307 Hz) are observed in the force spectra. The vortex structures are also in 2S mode. When Re =9.50 × 10⁴ (U_r = 10.51), the oscillation is still in the upper branch but tends to enter the transition branch, a prominent peak (f_osc = 0.868) and a small peak (f_osc = 1.853) in the displacement spectra, and the second harmonic appears in the spectrum, indicating that the vortex shedding mode of the fluid changes to 2P mode, that is, 2 pairs of vortexes fall off from the oscillator during one oscillation period. Additionally, 3 peaks (f_L = 0.918 Hz, f_L = 1.853 Hz, and f_L = 2.788 Hz) are observed in the force spectra.

For the spectral character of other diameters, as D = 0.12 m at Re = 5.79 × 10⁴ (U_r = 4.44), D = 0.14 m at Re = 6.78 × 10⁴ (U_r = 3.81), and D = 0.16 m at Re = 7.71 × 10⁴ (U_r = 3.33), these reduced velocities are relatively small, the vortex shedding cannot be induced or kept stable, and no prominent peak is observed in the force spectra. For D = 0.16 m at Re = 9.86 × 10⁴ (U_r = 4.26), a prominent peak of f_osc = 0.651 Hz is observed in the displacement spectra, and two peaks are observed in the force spectra (f_L = 0.651 Hz and f_L = 1.319 Hz).

While D = 0.12 m at Re = 9.40 × 10⁴ (U_r = 7.22), there is a prominent peak (f_osc = 0.818) at the first harmonic component, another peak is observed in the displacement spectra (f_osc = 1.653 Hz), and two prominent peaks are observed in the force spectra (f_L = 0.818 Hz and f_L = 1.653 Hz). For D = 0.12 m at Re = 11.41 × 10⁴ (U_r = 8.76), there are two peaks at the oscillating frequency (f_osc = 0.868 Hz and f_osc = 1.736 Hz), and three peaks in the force spectra are observed (f_L = 0.868 Hz, f_L = 1.736 Hz, and f_L = 2.604 Hz). As D = 0.14 m, both Re = 10.97 × 10⁴ (U_r = 6.19) (f_osc = 0.751 Hz and f_osc = 1.502 Hz) and Re = 13.31 × 10⁴ (U_r = 7.51) (f_osc = 0.785 Hz and f_osc = 1.553 Hz) have two peaks in the displacement spectra. While Re = 10.97 × 10⁴ (U_r = 6.19), four peaks are observed in the force spectra (f_L = 0.751 Hz, f_L = 1.503 Hz, f_L = 2.254 Hz, and f_L = 3.005 Hz), and the force spectra have two peaks when U_r = 7.51, which are f_L = 0.785 Hz and f_L = 1.553 Hz, respectively. As D = 0.16 m at Re = 15.21 × 10⁴ (U_r = 6.57), in the displacement spectra, two peaks are f_osc = 0.735 Hz and f_osc = 1.486 Hz, respectively, and four peaks are observed in the force spectra (f_L = 0.735 Hz, f_L = 1.486 Hz, f_L = 2.220 Hz and f_L = 3.706 Hz). These results indicate that all of the vortex modes are 2P.

3.4. Energy Conversion Characteristicsr

To study the effect of Re on the levels of energy conversion, the active power versus current velocity (U), reduced velocity (Ur), and Reynolds number (Re) for different ζ_total values for six oscillators are shown in Figure 18a–e. The power outputs are at six Reynolds number ranges, i.e., Re = 28.9 k~65.1 k (D = 0.06 m) and Re = 38.6 k ~ 86.7 k (D = 0.08 m) in the regime of TrSL2, TrSL3; Re = 48.2 k ~95.0 k (D = 0.10 m) in the regime of TrSL3; Re = 57.9 k ~114.1 k (D = 0.12 m); Re = 67.5 k~133.1 k (D = 0.14 m); and Re = 77.1 k ~ 152.1 k (D = 0.16 m) in the regime of TrSL3 and TrBL0.

It is worth noting that the Reynolds number affects not only the value but also the overall energy conversion. The maximum active power of D = 0.06 m is about 1.531 W (ζ_total = 0.122, Re = 6.51 × 10⁴, U_r = 19.98) and D = 0.08 m is 4.09 W(ζ_total = 0.152, Re = 8.67 × 10⁴, U_r = 14.99), while that at D = 0.10 m, D = 0.12 m, D = 0.14 m and D = 0.16 m are about 6.85 W(ζ_total = 0.246, Re = 9.50 × 10⁴, U_r = 10.51), 7.21 W (ζ_total = 0.197, Re = 11.40 × 10⁴, U_r = 8.76), 9.66 W(ζ_total = 0.737, Re = 12.84 × 10⁴, U_r = 7.24), and 10.43 W(ζ_total = 0.468, Re = 14.67 × 10⁴, U_r = 6.34), respectively. Similar to the corresponding vibration responses for six different Reynolds number ranges given in Figure 5, as Re increases, the active power shows an increasing trend in the VIV upper branch. However, when the oscillation goes into the VIV-Galloping branch, the active power output tends to decrease with the increase of Re. It is worth mentioning that when D is over 0.14 m, the oscillation stability and intensity are affected by large deformations of the water surface, but the maximum power output of D = 0.16 m is still larger than that of D = 0.14 m. For the same Reynolds numbers range cases investigated, the FIMECS generates more power as the system total damping ratio ζ_total is higher in the VIV upper branch.

3.5. The Upper Limit of Power Output

In this section, the influence of Re on the upper limit of power output P_UL is discussed. Figure 19 presents the variation of P_UL as a function of velocity at the maximum system total damping ζ_total,max of different oscillators can overcome.

For D = 0.06 m, D = 0.08 m, and D = 0.10 m, the upper limit of harvested power output P_UL increases continuously with the Reynolds number Re increasing. While for D = 0.12 m, D = 0.14 m, and D = 0.16 m, the P_UL increases with the Re until an optimal value and then decreases. This inconsistency may be because of the vibration response in different branches. Referring to Section 3.1, in the present experiment condition, for D = 0.06 with ζ_total = 0.122, the oscillation undergoes VIV branches and VIV-Galloping transition branches, and its maximum P_UL reaches 8.75 W, appearing at Re = 6.51 × 10⁴. For D = 0.08 with ζ_total = 0.152, the oscillation can be induced on a galloping branch, and its maximum P_UL reaches 13.25 W, appearing at Re = 16.22 × 10⁴. For D = 0.10 and ζ_total = 0.246, the VIV lower branch cannot enter the galloping branch. Its maximum P_UL reaches 15.27 W appearing at Re = 9.51 × 10⁴. As D = 0.12 with ζ_total = 0.423, D = 0.14 with ζ_total = 1.158 and D = 0.16 with ζ_total = 0.678, all of the oscillations enter into VIV branches. Their maximum P_UL are 12.83 W (Re = 9.80 × 10⁴), 14.73 W (Re = 13.31 × 10⁴) and 17.80 W (Re = 15.21 × 10⁴) respectively. It is noted that even the optimal total damping ratio of D = 0.14 m is higher than that of D = 0.16 m, but the P_UL of D = 0.16 m is larger than that of D = 0.14 m. The larger the characteristic width D of the oscillator, the higher the generating power of FIMECS. It can be seen that increasing the Re number range of the oscillator is conducive to improving the capacity of a single unit of a FIMECS. This is because the larger the Re number is, the larger the maximum system damping ratio that can be overcome by the oscillator is, and the larger system damping ratio is conducive to improving the generating power of the FIMECS.

The effect of Re on the upper limit of power output efficiency is also clearly evaluated in Figure 20. The η_UL first increases with the velocity and then decreases. While the change of η_UL with the Reynolds number is not regularly, these results seem to be due to the power in the current flow P_w varying with the cube of velocity U³. More power does not mean high efficiency. As derived in Equations (11) and (12), when the power available in the flow P_w dominates the variation of efficiency η_UL, even if P_UL undergoes a steady increase, η_UL may remain unchanged or even decline. In our test, the maximum η_UL of Cir-T-Att reaches 42.11%, appearing at D = 0.1 m, ζ_total = 0.246, and Re = 6.49(U_r = 7.18, U = 0.74 m/s).

$${\eta }_{\mathrm{UL}}(\%)=\frac{P_{\mathrm{UL}}}{P_{\mathrm{W}}}\times 100$$

(11)

$$P_w=\frac{1}{2}\rho U^3(2A_{\max }+D)\cdot l$$

(12)

4. Conclusions

All experiments are performed in the free surface water channel to investigate the influence on energy harnessing from flow-induced motion with the Cir-T-Att oscillator under six Reynolds number ranges (TrSL2, TrSL3, and TrBL0 flow regimes). We applied the controllable magnetic damping system to change the ζ_total by varying excitation voltages (V_B) and introduced a torque sensor to calculate fluid force. The V_B ranges from 0 V to 165 V, corresponding to ζ_total changes from 0.082 to 1.153. Some important conclusions were given as follows:

(1)

As Re increases, the oscillation intensity strengthens, and the range of the upper branch widens. Additionally, the maximum system total damping ζ_total,max of different oscillators can overcome increases;

(2)

In our experiments, with the increase of Re, when the oscillation goes into VIV branches, C_L shows an increasing trend, but as Re keeps on increasing, C_L shows a decreasing trend when the oscillator enters into VIV-Galloping transition branches;

(3)

In our measurements, compared with the higher harmonic component in the force spectrum, that of the displacement spectrum has less influence;

(4)

Both the global oscillation response and the levels of energy conversion are affected by the Re, the maximum P_harn of Cir-T-Att reaches 10.43 W, appearing at D = 0.16 m with Re = 1.47 × 10⁵(ζ_total = 0.468, U_r = 6.34, and U = 1.04 m/s);

(5)

As Re increases, the upper limit of harvested power P_UL increases in the VIV upper branch, and the maximum P_UL of Cir-T-Att reaches 17.80 W, appearing at D = 0.16 m with Re = 1.41 × 10⁵ (ζ_total = 0.678, U_r = 6.11, and U = 1.01 m/s).